His research was mainly focused on the theory of complex functions (meromorphic functions, univalent functions, invariants of analitic extension) and on differential geometry and algebraic topology, especially on knot theory (among others results, we mention the Călugăreanu theorem and invariant). He was an initiator of teaching the theory of complex functions and contributed to the field through a treatise published by the Didactic and Pedagogic Publishing House (1963, Bucharest).

Life and career

Călugăreanu Gheorghe was born on July 16, 1902, in Iasi (Romania), in a family of distinguished scholars. He attended primary school in Bucharest between 1909 and 1913 and secondary school at the renowned high school “Gh. Lazar“ (Bucharest) between 1913-1921.

In high school he became interested in natural sciences and so in 1921 he enrolled on the courses of the faculty of mathematics and physics at the University of Cluj, which he graduated in 1924. His relocation to Cluj can be explained by a larger historical context.

Following the Union of Transylvania with Romania on December 1, 1918, the question of organizing higher education in Romanian Transylvania was posed. In July 1919, a university commission (chaired by prof. Sextil Puşcariu) was constituted in order to organize the University of Cluj. This commission included: N. Iorga, V. Babeş, V. Pârvan, D. Gusti, Gh. Ţițeica, Gh. Marinescu, P. Poni. The university began to operate through four departments: Philosophy and Letters, Sciences, Law and Medicine. The Faculty of Sciences was to be divided into four sections: Mathematics, Physical – Chemical Sciences, Natural Sciences and Geography. On October 1, 1919, the University of Cluj, until then called Franz Joseph University became a Romanian institution, called Superior Dacia University “King Ferdinand I”. It was the third university in Romania where courses were taught in Romanian, after Iasi and Bucharest. Dimitrie Călugăreanu (father of Gheorghe) was appointed professor of Animal Physiology at the Faculty of Sciences in Cluj, with the task of organizing physiological education. He became a full professor there in January 1920. Professor Sextil Puşcariu became the first rector of the University of Cluj. Dimitrie Călugăreanu was elected dean of the Faculty of Sciences with Alexandru Borza as vice-dean, both professors at the Department of Natural Sciences. V. Babes and E. Racoviţă joined S. Puscariu and moved to Cluj, whilst others commuted: N. Iorga, D. Pompeiu, V. Pârvan, C. Levaditi. In 1920, D. Călugăreanu succeeded S. Puşcariu as Rector of the University of Cluj.

In 1922, whilst still a student, Gheorghe Călugăreanu was appointed assistant at the Institute of Theoretical and Applied Physics at the University of Cluj. In 1924 he graduated from the Faculty of Science with a specialization in Mathematics and a thesis in integral equations, one of the most modern chapters of mathematics at that time.

In 1926 he went to Paris as a Romanian state Fellow and attended courses taught by some of the greatest mathematicians of the era (Emile Picard, Jacques Hadamard, Élie Cartan, Paul Montel, Arnaud Denjoy and Gaston Julia). The same year, he received a degree from the Sciences University (Sorbonne) in Paris and in 1928 he was awarded the Ph. Doctorate in mathematical sciences at the same university. His dissertation “Sur les fonctions polygenes d’une variable complexe” (advisor Emile Picard), made important contributions to the study of polygenic functions, a field initiated by the great Romanian mathematician Dimitrie Pompeiu.

Back in Romania, he worked as an assistant (1930-1934), an associate professor (1934-1942) and from 1942, a professor at the University of Cluj. Here, he put all his talent and energy into the development of education and science in Romania and did so until the last moments of his life.

Between 1940-1945, several changes occurred at the University due to the second World War. In 1940, after the beginning of the war, the Hungarian University of Szeged moved back to Cluj and the Romanian University of Cluj moved to Sibiu and Timisoara (where Călugăreanu spent the war). In 1945, after the war’s end, the Romanian university returned to Cluj taking the name Babeș University. A part of the Hungarian University returned to Szeged, the other stayed at Cluj, named Bolyai University. Eventually the two merged under the name of Babeș – Bolyai University (1959).

Due to his teaching skills and scholarly work, Gh. Călugăreanu became one of the most esteemed professors of the University of Cluj, strengthening the prestige of the school of the theory of functions and topology. His courses were carefully prepared and presented with exquisite clarity. They became a model for several generations of mathematicians, who developed under his generous direction for nearly half a century. At the same time, Gheorghe Călugăreanu played an important role in establishing and streamlining the teaching of mathematics at Cluj University, as dean of the Faculty of Mathematics and Physics (1953-1957) and as head of the Department of Theory of Functions.

He wrote a course of the theory of analytical functions (lithographed), later turned into an excellent manual of theory of complex functions (Didactic and Pedagogic Publishing House, Bucharest, 1963). In collaboration with Professor D.V. Ionescu, he wrote a valuable manual of Calculus (2 volumes, lithographed). In 1964 he was awarded the title of “professor emeritus” for his high pedagogical mastery. In recognition of his scientific merit, in 1955 he was elected a corresponding member of the Romanian Academy, and in 1963 he became a member of the Romanian Academy.

Gheorghe Călugăreanu had an extensive and thorough knowledge of the field of mathematics, but he also had a wide cultural horizon. He graduated the Bucharest Conservatory (class teacher Florica Muzicescu) at the same time he graduated high school. He was an excellent pianist and a passionate music lover. His creative talent and scientific force were complemented by an outstanding artistic sensibility. Moments of tension and spiritual satisfaction engendered by his research were followed by those sublime moments when the scientist sat at the piano and became a true artist.

Throughout his career, he guided a great number of students, the most distinguished being Petru Mocanu, a corresponding member of the Romanian Academy, who succeeded him as the head of the Department of Function Theory at the Faculty of Mathematics (Babes-Bolyai University, Cluj).

An imposing stature, Călugăreanu had an unassuming character. He was gentle but demanding, a deep thinker, measured in his actions and a paragon of moral conduct. He had a full and happy family life with his wife Zoe (b. Filodor, married in 1941) and their two children, who continued the family tradition and became a biologist (Maria Luiza Flonta, b.1944 Timișoara) and a mathematician (Grigore Dumitru Călugăreanu, b. 1947 in Cluj).

Professor and Academy member Gheorghe Călugăreanu was awarded several Romanian orders and medals: 1953, the Medal of Labour; 1962, the Order of Labor, Class II; 1966, the Order of Scientific Merit class 1; 1969, the Medal “25th anniversary of the liberation of the country”; 1972, the Medal “25 years after proclamation of the Republic”, 1974, the Medal “30 years after the liberation”. An issue of the Revue Roumaine de Mathematiques Pures et Appliques was dedicated to him (vol. 17, no. 9, 1972), to celebrate his 70th birthday.

On November 15, 1976, still driven by his creative force, Călugăreanu Gheorghe passed away from cancer. Following his wish, he was cremated and the urn was deposited at Belu Cemetery in Bucharest.

Scientific work

The focus of Gheorghe Călugăreanu’s scientific work was the study of several fundamental problems of the theory of complex functions, geometry, algebra and topology. Continuing the tradition of his great predecessor Dimitrie Pompeiu, Călugăreanu began his research work with several valuable original contributions to the theory of complex functions. The focus of his PhD thesis and published papers from 1928, was the theory of polygenic complex functions . The study of these functions was initiated by Dimitrie Pompeiu, who introduced the areolar derivative to the theory of polygenic complex functions. This concept has subsequently found important applications in geometry, mechanics and mathematical physics. Starting from the observation that the areolar derivative coincides with the partial derivative of the function with respect to the conjugate of the independent variable, Gh. Călugăreanu studied, for the first time, the problem of the polygenic solutions of differential analytical equations. In his PhD thesis, he showed that there are classes of differential equations admitting polygenic solutions that are easier to obtain than monogenic solutions. He established a simple connection between these two types of solutions that makes it possible to form families of monogenic solutions of the equation.

From 1929 he started studying meromorphic functions. He focused on several problems related to Picard’s theorem and its generalizations. So he explored relations that exist between exceptional values in the sense of Picard and the sequence of the Taylorian coefficients of an element of the meromorphic function. He determined that in the case of finite genus p meromorphic functions, the possible exceptional values are given by an algebraic equation of degree p + 1, in which only the first p + 2 coefficients of the Taylorian element appear as well as the sums of certain series formed with the poles of the function. This result, first proved with R. Nevanlinna’s theory and later proved in an elementary way, led him to study other species of exceptional values, in the sense of Borel, Valiron and Nevanlinna. This way he obtained a generalization of the theorems on exceptional values, which also constitutes the definition of a new type of exceptional values (which Valiron calls exceptional values C).

The study of univalent functions is one of the central concerns of current research in geometric theory of analytical functions. Gh. Călugăreanu addressed a fundamental problem in this area of research. Through various methods, he searched for necessary and sufficient conditions for the univalence of an analytical function, inside or outside the unit disk. As a first result (1932) he found the univalence radius of the function, which coincides with the principal radius of convergence of a double series formed using the Taylorian series of the function. Thus he gave for the first time, necessary and sufficient conditions of univalence, which he wrote in different forms. He came back to this problem later, in a series of articles published between 1950-1965. In 1954, using a method based on a generalization of the area principle, he obtained infinitely many necessary and sufficient univalence conditions on the Taylorian coefficients of the function. Using another ingenious method, based on the introduction of some singular integral, necessary and sufficient conditions of integral form are obtained even in the more general case of univalent domains in an arbitrary Euclidean space. Starting from the observation that the inverse of a uniform function is univalent in any disc where it is holomorphic, he gave a very simple necessary condition for uniformity of analytical functions in 1933.

The discovery of the invariants of analytical extension is one of the most important contributions of Călugăreanu to the theory of analytical functions. Building on previous research on meromorphic and on univalent functions, he tackled a more general problem, namely that of the relations between the qualitative or quantitative properties of an analytical function and the Taylorian coefficients of an element of this function. Noting that when it comes to global properties that view the function as a whole, expressing these properties by equalities or inequalities between coefficients of an element, is independent of the chosen element, he was able in 1936 to define invariants for extension of analytical functions. These are expressions formed with Taylorian coefficients of an element of an analytic function whose value does not change when an element is replaced with any other element of the same analytical function. Through a series of papers, he managed to build successively such invariants for different classes of analytic functions. He drew up the general theory of extension invariants in 1939 when noting that these expressions are actually analytical functionals (known in the literature as Călugăreanu functionals), and he used L. Pantappie analytical functional theory in order to obtain a complete system of extension invariants and covariants in the general case. The results he obtained shed a new light in the study of the global properties of analytic functions. The theory of extension invariants that is presented in the book by P. Levy (Problèmes concrets d’Analyse fonctionelle, Paris, 1961) was the subject of several papers of Fr. Pellegrino and of the thesis of Fr. Succi (1950).

Attempts to solve the problems raised by the theory of extension invariants using Chebyshev’s polynomials for complex numbers, led Călugăreanu to a series of important results. He obtained a remarkable special form of the Chebyshev’s polynomials of a compact set in the plane and generalizations of the transfinite diameter and their use in singularities problem.

Gh. Călugăreanu has also made important contributions in other areas of mathematics, namely geometry and topology. His first work in 1924, reveals a characteristic property of quadrics, relative to three arbitrary points of the surface that is implied by a law of electrostatic distribution on an conductor ellipsoid. He resumed this problem in 1964 in a more general form, obtaining multilocal differential very simple relationships that characterize algebraic curves. Another important problem of differential geometry that he studied was that of the intrinsic representation of surfaces by expressing mean curvature and total curvaturein terms of the geodesic coordinates. In 1942 he begun a study of singularities occurring in the pointwise transformations of a plane to another plane, establishing the existence of the folds and single or multiple vertices, and their absence in the case of conformal transformations and their topological equivalents. In 1941, in collaboration with Gh. Th. Gheorghiu, he obtained geometric interpretations of affine and projective differential invariants of plane curves.

The area of topology which particularly attracted Gh. Călugăreanu was knot theory. He worked passionately in this area from 1942 until the last moments of his life. As in other areas of his research, he tried to solve one of the crucial problems of the theory, namely that of the characterization of knots in terms of isotopy, seeking a complete system of topological invariants that characterize the isotopy classes of knots. The first invariants he obtained successively in 1942, 1959 and 1961 were given in integral form (of Gauss integral type). One of these invariants was resumed by the American mathematicians W.F. Pohl (1968), J.H. White (1969 thesis) and F. Brock Fuller, who applied the invariant to the problem of the twisting of DNA molecules, a problem of interest in molecular biology (1971).

A quote from “Helicity and Călugăreanu’s invariant” (see bibliography): ‘Curiously, Fuller (1972), in an article dedicated to Călugăreanu (the anniversary of 70 years), attributed to White (1969) the result n = W + Tw although this result with Tw = T + N can be found clearly spelled out, and coupled with an analysis of the role of inflection points in Călugăreanu’s article (1961). White’s achievement is to place this result in the wider context of arbitrary dimension differential manifolds; but this theorem in the form n = W + T + N, or the equivalent form n = W + Tw should undoubtedly described as Călugăreanu’s theorem. We feel the need to emphasize this because in recent articles and books, Călugăreanu is less than should be credited for its completion. Thus, for example Pohl (1980) describes the above formula as the formula of White and almost reluctantly (sic) states that “the formula of White was presented actually by Georges Călugăreanu (1961), initially for curves with the curvature nowhere zero. This demonstration was very difficult and its formulation, confused.” We question this judgment and remember that (1961) Călugăreanu explicitly considered the problem of zero or inflectional curvature, while (1969) White explicitly excluded these considerations. An overall misunderstanding of Călugăreanu contribution gradually led the world of mathematics to refer to the above formulas as White’s theorem, so that even in textbooks (e.g., Kauffman, 1987, p. 18; 1991, p. 489) references to Călugăreanu’s articles 1959, 1961, disappeared little by little. ‘

A more detailed study of the operation of crossing (1962) and the methods of generating knots (1965) allowed him to reduce the problem to a two dimensional problem, that of classifying simple closed curves drawn on a closed orientable surface which separate the surface into two disjoint domains, which amounts to determine some elements of finitely generated Fuchs groups. In 1966-68 he gave the criteria for the recognition of these elements. In order to form a complete system of isotopy invariants for three-dimensional knots, he was led to define and form a system of invariants of contraction in a group given by generators and relations (1970-1971). In 1975 he gave a geometric proof of a theorem of M.H. Zieschang. In his last paper (1976, 48 pages!) he made a comprehensive study of some invariants associated to countable groups.

His focus on the discovery of invariants, which is the red thread throughout his work, has the origin in his permanent aspiration to capture what is durable and characterizes a certain mathematical entity. This belief has been exposed with great clarity and skilfulness in his festive lecture given in 1972, held on the occasion of his retirement.

Characterizations by scientists who knew him during his life

Peter Mocanu (1992):His mathematical work remains a model of elevated and unitary spiritual construction.His work places Gheorghe Călugăreanu in the gallery of the most distinguished representatives of the Romanian school of mathematics.

Caius Iacob (1992):Călugăreanu’s rich scientific work, the importance of which is becoming increasingly evident as time passes, is characterized by a harmonious structure, welded with extreme elegance and depth.Uninfluenced by passing trends or transient fashions in mathematics, Călugăreanu, a strong personality, followed an original and specific path in his scientific work, established early in his life.

Ionel Maftei (1978):Those who knew him remember professor Călugăreanu with his ever-present cigar, the gentle expression in the eye, sometimes slightly ironic, but always sympathetic.He was quiet and reserved, never talked about himself even with those closest to him.Scientific and academic accomplishments never affected his behaviour or attitude towards his peers.Vanity, arrogance or complacency were always foreign to him.He seemed to notice his failures in his research more than his successes. He was always kind and had praise for the successes of others. What drove him was his passion for mathematics and the awareness of being useful.He talked about his own life only insofar as he believed that it contained some useful lessons for the new generations.He had a full life and knew how to enjoy even the most ordinary moments of life. He was focused when teaching and conducting, relaxed in front of the piano and with his family, together with his wife and his children.